226 CHAPTER 5 Discrete Probability Distributions 0.00165855 is not the probability of getting exactly two adults who are cashless because it was found by assuming a particular sequence. Other different sequences are possible. In Section 4-4 we saw that with two subjects identical to each other (such as adults who are cashless) and eight other subjects identical to each other (such as adults who are not cashless), the total number of arrangements, or permutations, is 10!>3110 - 22! 2!4 or 45. Each of those 45 different arrangements has a probability of 0.052 # 0.958, so the total probability is as follows: P12 adults are cashless among 102 = 10! 110 - 22!2! # 0.052 # 0.958 = 0.0746 This particular result can be generalized as the binomial probability formula (Formula 5-5). That is, the binomial probability formula is a combination of the multiplication rule of probability and the counting rule for the number of arrangements of n items when x of them are identical to each other and the other n - x are identical to each other. P1x2 = n! 1n - x2!x! # px # qn-x 2 2 Binomial Distributions Access tech supplements, videos, and data sets at www.TriolaStats.com TECH CENTER Minitab 1. Enter the values of x for which you want probabilities (such as 0, 1, 2, 3, 4, 5) in column C1. 2. Select Calc from the top menu. 3. Select Probability Distributions from the dropdown menu and Binomial from the submenu. 4. Select Probability, enter the number of trials, enter the event probability, and select C1 for Input Column. 5. Click OK. StatCrunch 1. Click Stat in the top menu. 2. Select Calculators from the dropdown menu and Binomial from the submenu. 3. In the dialog box, enter the desired values for n, p, x. Select =or the desired inequality for x. 4. Click Compute. Excel 1. Enter the values of x for which you want probabilities (such as 0, 1, 2, 3, 4, 5) in column A. 2. Select cell B1, click Insert Function ƒx, select the category Statistical, select the function BINOM.DIST and click OK. 3. Enter A1 for Number_s and then enter the number of trials n and probability p. 4. Enter 0 in the Cumulative box. 5. Click OK and the probability will appear in cell B1. 6. Copy B1 down the column to obtain the probability for each value of x listed in column A. Tip: Enter 1in Step 4 for the cumulative binomial distribution. R R command: dbinom(x,n,p) TIP: Use the R command pbinom(x,n,p) for cumulative probabilities. A complete list of R statistical commands is available at TriolaStats.com The number of outcomes with exactly x successes among n trials The probability of x successes among n trials for any one particular order Statdisk 1. Click Analysis in the top menu. 2. Select Probability Distributions from the dropdown menu and select Binomial Distribution from the submenu. 3. Enter the values for n, p and click Evaluate. Tip: Enter a specific value for x to get a single probability. TI-83>84 Plus Calculator 1. Press F then O keys to access the DISTR (distributions) menu. 2. Select binompdf and click [. 3. Enter the values for trials n, probability p, and number of successes x to complete the command binompdf(n, p, x). Press [. Tip: Omitting a value for x provides a list for all probabilities corresponding to x = 0, 1, 2…n. Press Y then F then 2 to save the probabilities as list L2. You can then manually enter the values of x in list L1 for calculations. Tip: Select binomcdf in Step 2 for cumulative probabilities.
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